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Displaying similar documents to “Dimension of the harmonic measure of non-homogeneous Cantor sets”

p Harmonic Measure in Simply Connected Domains

John L. Lewis, Kaj Nyström, Pietro Poggi-Corradini (2011)

Annales de l’institut Fourier

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Let Ω be a bounded simply connected domain in the complex plane, ℂ . Let N be a neighborhood of ∂ Ω , let p be fixed, 1 < p < ∞ , and let u ^ be a positive weak solution to the p Laplace equation in Ω ∩ N . Assume that u ^ has zero boundary values on ∂ Ω in the Sobolev sense and extend u ^ to N ∖ Ω by putting u ^ ≡ 0 on N ∖ Ω . Then there exists a positive finite Borel measure μ ^ on ℂ with support contained in ∂ Ω and such that ∫ | ∇ u ^ | p - 2 〈 ∇ u ^ , ∇ φ 〉 d A = - ∫ φ d μ ^ whenever φ ∈ C 0 ∞ ( N ) . If p = 2 and if u ^ is the Green function for Ω with pole at x ∈ Ω ∖ N ¯ then the...

Doubling conditions for harmonic measure in John domains

Hiroaki Aikawa, Kentaro Hirata (2008)

Annales de l’institut Fourier

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We introduce new classes of domains, semi-uniform domains and inner semi-uniform domains. Both of them are intermediate between the class of John domains and the class of uniform domains. Under the capacity density condition, we show that the harmonic measure of a John domain D satisfies certain doubling conditions if and only if D is a semi-uniform domain or an inner semi-uniform domain.

Multifractals and projections.

Fadhila Bahroun, Imen Bhouri (2006)

Extracta Mathematicae

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In this paper, we generalize the result of Hunt and Kaloshin [5] about the L-spectral dimensions of a measure and that of its projections. The results we obtain, allow to study an untreated case in their work and to find a relationship between the multifractal spectrum of a measure and that of its projections.

Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Matthew H. Baker, Robert Rumely (2006)

Annales de l’institut Fourier

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Given a rational function ϕ ( T ) on ℙ 1 of degree at least 2 with coefficients in a number field k , we show that for each place v of k , there is a unique probability measure μ ϕ , v on the Berkovich space ℙ Berk , v 1 / ℂ v such that if { z n } is a sequence of points in ℙ 1 ( k ¯ ) whose ϕ -canonical heights tend to zero, then the z n ’s and their Gal ( k ¯ / k ) -conjugates are equidistributed with respect to μ ϕ , v . The proof uses a polynomial lift F ( x , y ) = ( F 1 ( x , y ) , F 2 ( x , y ) ) of ϕ to construct a two-variable Arakelov-Green’s function g ϕ , v ( x , y ) for each v . The measure μ ϕ , v is...